Variance Practice Questions - NDA Mathematics Question Bank

Question 1 of 5 NDA MCQ

The variance of five positive observations is 3.6. If four of the observations are 2, 2, 4, 5 then what is the remaining observation?

Detailed Explanation

Explanation:

Let x be the 5th observation

∴ Variance = (2² + 2² + 4² + 5² + x²)/5 − ((2 + 2 + 4 + 5 + x)/5)²

3.6 = (49 + x²)/5 − (13 + x)²/25

⇒ 3.6 × 25 = 5(49 + x²) − (13 + x)²

⇒ 90 = 245 + 5x² − 169 − x² − 26x

⇒ 2x² − 13x − 7 = 0

⇒ x = 7, x = −1/2

(∵ observation is positive)

∴ x = 7

Question 2 of 5 NDA MCQ

If two random variables X and Y are connected by relation 2X-3Y/5X+4Y = 4 and X follows Binomial distribution with parameters n = 10 and p = 1/2 , then what is the variance of Y?

Detailed Explanation

Question 3 of 5 NDA MCQ

Consider the following for the next two (02) items that follow:
The marks obtained by 10 students in a Statistics
test are 24, 47, 18, 32, 19, 15, 21, 35, 50 and 41

What is the variance of the largest five observations?

Detailed Explanation

Question 4 of 5 NDA MCQ

Direction for Questions (95-96): Consider the following for the two (02) items that follow: The sum and the sum of squares of the observation corresponding to length X (in cm) and weight Y (in gm) of 50 tropical tubers are given as ∑X = 200, ∑Y = 250, ∑X² = 900 and ∑Y² = 1400.

Which of the following is correct?

Detailed Explanation

Question 5 of 5 NDA MCQ

If the random variable X has mean 3 and standard deviation 5, then what is the variance of the random variable

Y = 2X − 5 ?

Detailed Explanation

Solution

Option (d) is correct.

Explanation:

\[ \text{Variance of } Y = 2X - 5 \]

\[ \text{Var}(Y) = \text{Var}(2X - 5) \]

\[ \text{Var}(Y) = 2^2 \cdot \text{Var}(X) \]

\[ \text{Var}(Y) = 4 \cdot \text{Var}(X) \]

\[ \text{Var}(X) = \sigma^2 = 5^2 = 25 \]

\[ \text{Var}(Y) = 4 \cdot 25 = 100 \]

📌 Hints / Properties Used:

Variance of a linear transformation: Var(aX + b) = a^2 * Var(X).
Standard deviation and variance relationship: \(\sigma = \sqrt{\text{Var}(X)}\).

📊 Visual Diagram Suggestion:

A flowchart illustrating the transformation of the random variable X to Y, showing how mean and variance transform under linear transformations.

📖 Factual Verification & Reference:

Verified against standard statistical principles (e.g., NCERT Class XI Mathematics, Chapter on Statistics).

Variance

Solution

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